Suppose that X is a continuous random variable whose probability density function is given by: #f(x)= k(2x - x^2) for 0 < x < 2#; 0 for all other x. What is the value of k, P(X>1), E(X) and Var(X)?

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Answer 1 · 2016-11-13T08:31:00

#k=3/4#
#P(x>1)=1/2#
#E(X)=1#
#V(X)=1/5#

To find #k# , we use #int_0^2f(x)dx=int_0^2k(2x-x^2)dx=1#

#:. k[2x^2/2-x^3/3]_0^2=1 #

#k(4-8/3)=1# #=>##4/3k=1##=>##k=3/4#

To calculate #P(x>1)# , we use #P(X>1)=1-P(0< x <1) #

#=1-int_0^1(3/4)(2x-x^2)=1-3/4[2x^2/2-x^3/3]_0^1#

#=1-3/4(1-1/3)=1-1/2=1/2#

To calculate #E(X)#

#E(X)=int_0^2xf(x)dx=int_0^2(3/4)(2x^2-x^3)dx#

#=3/4[2x^3/3-x^4/4]_0^2=3/4(16/3-16/4)=3/4*16/12=1#

To calculate #V(X)#

#V(X)=E(X^2)-(E(X))^2=E(X^2)-1#

#E(X^2)=int_0^2x^2f(x)dx=int_0^2(3/4)(2x^3-x^4)dx#

#=3/4[2x^4/4-x^5/5]_0^2=3/4(8-32/5)=6/5#

#:.V(X)=6/5-1=1/5#